I nternat. J. Math. Math. Si.
Vol. 4 No. 4 (1981) 745-752
745
ON THE ALMOST SURE CONVERGENCE OF WEIGHTED
SUMS OF RANDOM ELEMENTS IN D[0,1]
R.L. TAYLOR and C.A. CALHOUN
Department of Mathematics and Statistics
University of South Carolina
Columbia, S. C. 29208
(Received June 26, 1980 and in revised form February 17, 1981)
ABSTRACT.
W
Let {w } be a sequence of positive constants and W
n
n
n_Wn
and w /
/
n
D[O,1].
.
Let
{Wn }
Wl+...+ wn where
be a sequence of independent random elements in
-1
The almost sure convergence of W
n
n
k=l
Wk
is established under certain
integral conditions and growth conditions on the weights {w }.
n
The results are
shown to be substantially stronger than the weighted sums convergence results of
Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963)
and Daffer and Taylor (1979).
KEY WORDS AND PHRASES. Weighted Sums, Random Elements in D[O, I], Strong Laws of
Lge Numbr -ntegral Conditions, and Almost Sure Convergence.
1980 THEMATICS SUBJECT CLASSIFICATION CODES.
i
Primay 60B12; Secondary 60F15.
INTRODUCTION
The convergence of weighted sums of random elements in
D[0,1] was obtained by
Taylor and Daffer (1980) using a number of conditions such as convex tightness,
moment conditions, and others.
In this paper both the moment conditions and the
tightness conditions are substantially relaxed in obtaining the almost sure con-
vergence of weighted sums of random elements in D[0,1].
In addition, a brief
discussion of the necessary integral conditions will be included which will delineate these results and previous work.
746
R. L. TAYLOR AND C. A. CALHOUN
Let D[0,1](--D) denote the space of real-valued functions on [0,i] which
are right continuous and possess left-hand limits for each t
[0,i].
Let the
linear space D be equipped with the topology generated by the Skorohod metric d
and let
let
X:
ogy.
II II
denote the uniform norm,
(, A, P) denote a probability space.
/
llxll
sup
Ix(t)
for x
D.
Next,
t[0,1]
A random element X in D
is a function
D which is measurable with respect to the Borel sets of the Skorohod topol-
Random elements in D are characterized by the property that X(t) is a random
variable for each t E
by (EX)(t)
[0,I].
The expected value EX
m(x(t)) for each tE [0,I] when E
llXll
D can be defined pointwise
<
.
Detailed geometric and
probabilistic properties of the space D with the uniform norm and the Skorohod
metric can be found in Billingsley
(1968), pp. 109-153 and Taylor (1978),
pp. 153-184.
2.
INTEGRAL CONDITIONS.
In obtaining the strong law of large numbers for independent, identically
distributed random elements in D, Ranga Rao (1963) used the following crucial
LEMMA i.
If X is a random element in D with E
there exists a partition 0
max
sup
l_<i<m
ti_l_<t, s<ti
< t
I <...<
ElX(t)
X(s)
t
O
tm
llxlloo
< oo, then for each
E
> 0
i of [0,i] such that
-< e.
(2.1)
In extending inequality (2.1) to a sequence of random elements and in trying
to obtain strong laws of large numbers for non-identically distributed random
elements, the following formulation of integral properties were motivated.
each of the following properties and in Lemma i, the partition
{tl,...
t
m
In
depends
on E.
DEFINITION.
A sequence of random elements {X
n
in D is said to satisfy (a)
property (RT) if for each e > 0 there exists a partition 0
of [0,13 such that
t
0< t I <...< tm
i
CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS
sup
n
max
0-<i<m-i
E[IX___(ti+l
O)
X
n
(t)I]
i
747
e
(2.2)
(b) property (mT) if for each e > 0 there exists a partition
0
t
o <
t
I <...< t m
i of [0,i] such that
E[
max
sup
sup
n 0_<im-I
ti_<t<ti+
IXn(t)
Xn(t i)
I]
<
e.
(2.3)
I
(c) property (T) if for each e > 0 there exists a (Skorohod) compact set K such
that
sup E
n
fIXn I[Xn K]II
< e.
Clearly, property (mT) implies property (RT).
random elements
{Xn }
with E
IIXII<
(2.4)
Also, identically distributed
satisfy (RT) obviously and satisfy (mT) by
Lemma 2 since (T) follows from the tightness of identical distributions.
LEMMA 2. If {Xn } satisfies property (RT) and property (T), then {Xn
satis-
fies property (mT).
Let e > 0 be gdven,
PROOF.
sup E
n
Since
Choose K (Skorohod) compact such that
K]II
fiXn I[X
n
for each x
E
x(s)
e/6.
(2.5)"
> 0 such that
K is compact, there exists
Ix(t)
<
<-
Ix(u-0)
x(s)
+ e/3
(2.6)
K whenever 0 s s S t < u < s + 6.
By (RT) choose a partition {t l,...,tm } of [0,I] such that
sup max
i
n
E[[X"(ti+ILL
0)
L-X"(ti) l]
Without loss of generality it can be assumed that
i
0,i
m-l.
Thus, from (2.6) and (2.5)
sup max E[
t
i
n
sup
i<t.<ti+l
IXn(t)
Xn(t i)
l]
ti+.
< g/3.
tol <
(2.7)
for all
748
R.L. TAYLOR AND C. A. CALHOUN,
< sup max E[
n
i
+ sup
n
max E[
i
< sup max
n
i
+ sup
sup
<t< t
ti-
sup
ti<t<ti+I
n
max
i
IXn(t) Xn(ti)]l[xn
01Xn(t)
E[(IX_(ti+
I,
E[211
nll
X
n
-< sup
i+l
0)
l[xn
E[IXn(ti+l-0)
X
n
(t)I I IX
n
Xn (ti)
K]
K]
K]
+ e/3) I Ix
n
K]
]
Xn(ti) II Ixn
K]
]
+ el3 + el3
III
by (2.7).
Property (T) has been used in several laws of large numbers for random elements in Banach spaces to eliminate the assumption of identical distributions.
In obtaining laws of large numbers for D, Daffer and Taylor (1980) required convex tightness or that the (Skorohod) compact set in (2.4) also be convex.
In
particular, the previous convex tightness condition can be partially formulated
by the integral condition
sup E [max
n
i
sup
ti-<t<ti+I
IXn(t)
Xn(t i)
l] <- e.
(2.8)
Property (mT) is clearly much more general than (2.8), and convergence results in
the next section using property (mT) are a marked improvement over the results of
Taylor and Daffer (1980) and include the results for identical distributions.
3.
ALMOST SURE CONVERGENCE.
Let { w
k}
W
n
w
I
+...+
variables
denote a sequence of positive constants.
w
n
S
n
{X } are said
n
Ink= I WkXk,
and N(x)
card {n:
Define
W-n
i
Wn
x}.
Random
to have uniformly bounded tail probabilities if there
exists a nonnegative real-valued random variable X such that
P[IXnl
by {Xn
> t] -< P[X > t]
c X.
for each n and for all t
R.
This property is denoted
The following real-valued result can be obtained from the geometric
CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS
749
Banach space results of Howell, Taylor and Woyczynski (to appear) and will be
used in establishing the almost sure convergence of weighted sums of random
elements in D.
Let {X } be independent (real-valued) random variables such that
THEOREM i.
n
{X } e X, EN(X) < =, and for some
n
f0
If W
n
/
,
t
P-I P[X > t]
i < p < 2
y-(P+l)
ft
=.
N(y)dy dt <
then
W-Is
-c /0.
n
n
n
a.s.
where
c
REMARK.
If
W-i
n
n
-i
Wn Wn
/
’
n
We m( l[iXk
’k=l
EX <
-i
Wn Sn EXI
,
/
and EX
_<
EXn
I
w
I
Wk ])
for all n, then
0
since
EX I
EX
n
n
[IXnl
< w
-I
n
;w-lW tdP[IXnl
W ]
n
n
tdP[IXnl
n
EXn
If W
/
n
EX
and
1
0
n
as
such that
n
P[X >
wn
n
]
f=ow-iW P[X >
+
t]dt
n
-+=#.
Let {X } be independent random elements in D such that
n
for all n, and
wn
/
n
=,
IIWn-I
PROOF.
n
n
(ii) {
(i) property (mT) is satisfied,
(iii)
w-
n
n
/
THEOREM 2.
> t] <
-< t]dt
IIXnll }
X and EN(X) <
(iv) for some i _< p _< 2
,
Condition (3.1) holds.
then
S
n
EX
Ill
/
0
a,s,
By property (mT) there exists a partition 0
t
O
< t
I
<...<
tm
i
R. L. TAYLOR AND C. A. CALHOUN
750
E[
max
O<i<m-i
sup
ti<t<ti+I
[Xk(t) Xk(ti) [3
e
(3.2)
for each k.
For each n
<
max
Oi<m-i
.
n
W-I
k=lWk
n
sup
ti<t<ti+I
l(t) (tl)
IWI [k=l Wk(tl)
+
max
Oim
+
max
OSim-i
n
sup
ti<t<ti+I
EXl(ti)
IEXl(t)
EX
l(t i) l.
(3.3)
Similar to the proofs in Taylor and Daffer (1980), the last term in (3.3) is less
than e by property (mT).
Also, noting that for each i
term in (3.3) converges to 0
a.s.
{(ti)}
X, the second
For the first term of (3.3),
by Theorem i.
define
ti<t<-ti+1
For each i {z_(i)} are independent random variables with zero means and
k
} a 2X. Thus
{zk(i)
(i)
W-I
n in
k=l WkZk
by Theorem i, for each i
lira sup
n
a.s.
0
0,1,...,m-l.
Wllk=In WkY i)
for each i by property (mT).
lim sup
n
/
a.s.
Also, by Theorem i
< lira sup
n
Wnl Ik=l Wk k(i) e
n
EY
(3.5)
Hence,
IIWn-llnk=l Wk- EXII
-<
a.s.,
and the proof is completed by taking a sequence of
cluding a countable union of null sets.
’s
(3.6)
going to zero and by ex-
///
CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENTS
751
To obtain the traditional strong law of large numbers, let wk
i for all k.
[y] (the greatest integer function), and condition (3.1) becomes
Then, N(y)
approximately
f0 tp-I P[X > t] ft y-(p+l)
for each p > i.
COROLLARY i.
Y dy dt
f0 P[X > t]dt
Thus, the following corollary is immediate.
independent
If {X } is a sequence of
n
that (i) property (mT) is satisfied
(li) {X }
n
random elements in D such
X and EX < =o, and (ill) EXn
EX I
then
lln -I I nk= I
It is easy to show that
(mr).
k
llXnl
I
I
n
fIXnll e
> t] <
COROLLARY 2.
fM_p s- (p+l) spM
n}
If {X
0
-< M for all n,
t-e M
will yield a random variable X such that
EX
/
EXn } has property (mT) if {Xn } has property
{X
Also, if for some p > I, g
P[
EX
X
f s- (p+l) pM
{X }
X
and
ds-- pM
_/M_p
n
s
then
ds
-p
ds <
.
is a sequence of independent random elements in D such
that (i) property (mr) holds and (ii) for some p > i, sup E
n
llXnll p
< ==, then
Corollary 2 represents a major improvement over Theorem i of Daffer and
Taylor (1979) in that not only is convergence in the Skorohod topology replaced
by convergence in the uniform norm but the restrictive condition of convex tightness is replaced by property (mT).
Finally, since identically distributed random
elements with a first absolute moment satisfy property
(mT), the following more
general form of the strong law of large numbers is available.
THEOREM 3.
If {X } are identically distributed random elements in D such
n
R. L. TAYLOR AND C. A. CALHOUN
752
that (i) EX
llXll
I
exists and EN(
llXll
<
and (ii) condition (3.1) holds for
X, then
n
I[W llk_-I
Wk EXI [[
/
0
aoS.
The authors are grateful to Peter Z. Daffer for his dis-
ACKNOWLEDGEMENTS.
cussions and help in the development of the integral conditions and their rela-
tionships, and to the referee for helpful comments on organization of the paper.
The research was supported in part by the Air Force Office of Scientific Research
under Contract number F49620-79-C-0140 and by the Research and Productive Scholarship Fund of the University of South Carolina.
.
REFERENCES
Billingsley, P.
Convergence of Probability Measures, Wiley, New York, (1968).
2.
Daffer, P. Z. and Taylor, R. L. Laws of large numbers for D[0,1], Annals
of Probabili.ty 7 (1979), 85-95.
3.
Howell, J., Taylor, R. L., and Woyczynski, W. A.
4.
Rao, R. R.
5.
Taylor, R. L.
Stability of linear forms
in independent random variables in Banach spaces, (to appear) Proc.
Prob. in Banac..h Spaces. III.
The law of large numbers for D[0,1]-valued random variables,
Theory of Prob. and Appl. _8 (1963), 70-74.
Linear
Convergence of Weighted Sums of Random Elements in
Lecture Notes in Mathematics, 672. Springer-Verlag,
Stochastic
Spaces.
Berlin, 1978.
6.
Taylor, R. L. and Daffer, P. Z. Convergence of weighted sums of random
elements in D[0,1], J. of Mult. Anal. i0 (1980), 95-106.